If you had a ridiculously bright laser, and you were positioned at a point far enough away, could you see the light move through space?

Question

Just a question out of pure curiosity that I can't seem to find an existing answer to. Let's say you are at an empty point in space, and your friend has the universes strongest and brightest laser 10 light seconds in front of you and to the left. If they fired the laser from your left to the right so that it crossed in front of you 10 light seconds away, would you be able to see the light moving through space?

Answer 1

Suppose you have a power $P$ beam of radius $r$, hitting dust of number density $n$ and radii $R$. Then each meter of beam would hit $N=\pi r^2 n$ dust particles, and they would each radiate $p=P \pi R^2$ Watt. If we simplify by assuming it is isotropic radiation that means that at distance $d$ we will see an irradiance of $I_1=Np/4\pi d^2 = \pi r^2 n P R^2/ 4 d^2$ Watt per square meter.

If we integrate this along an infinite beam at closest distance $d$ we get $I=(\pi r^2 n P R^2/ 4 )\int_{-\infty}^\infty \frac{dx}{d^2+x^2} = \pi^2 r^2 n P R^2 / 4 d$

If we set $N=10^{-15}$ per cubic centimetre, $R=30 \mu$, $d=10$ AU, $r=1$ m, the result is $I=1.5\cdot 10^{-30} P$. So for a solar luminosity beam ($10^{26}$ W) this will be $10^{-4}$ Watt per square meter. That sounds like it might actually be visible (briefly, as the dust turns into plasma).